Solvability Of Boundary Value Problems For A Class Of Fractional Elliptic Partial Differential Equations

By means of the saddle point theorem in the variational method,this paper mainly studies the solvability of boundary value problems for a class of fractional elliptic partial differential equations with double resonance.It is proved that under certain conditions,especially under the classical Ahmad-Lazer-Paul condition,there is at least one solution for this kind of boundary value problems.As a special case,the solvability of this kind of boundary value problem for fractional Laplace operator is given.In addition,this paper also explores the existence and multiple solutions of the nontrivial solutions of this kind of boundary value problems when some additional conditions are added to the nonlinear terms,and obtains the existence and multiple solutions of the nontrivial solutions of this kind of boundary value problems by using the knowledge related to the calculation of the critical groups in Morse theory,and also takes the fractional Laplace operator as a special case to prove the existence of its nontrivial solutions.